Finite-Volume Solution of the P3 Equations of Radiative Transfer and Coupling to Reacting Flow Calculations

2009 ◽  
Author(s):  
Mahesh Ravishankar ◽  
Sandip Mazumder ◽  
Ankan Kumar
Keyword(s):  
Radiative Transfer ◽  
Finite Volume ◽  
Unstructured Mesh ◽  
Laminar Flame ◽  
Cell Group ◽  
Reacting Flow ◽  
Computational Domain ◽  
Algebraic Equations ◽  
Participating Media ◽  
Cpu Time

The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled second-order partial differential equations. The FV procedure, presented here, is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the Binary Spatial Partitioning algorithm, followed by fully implicit coupled solution within each cell group using a pre-conditioned Generalized Minimum Residual (GMRES) solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for a benchmark problem. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the 4 P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the 10 equations that are solved for the reacting flow calculations.

Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes

2009 ◽  
Vol 132 (2) ◽  
Author(s):  
Mahesh Ravishankar ◽  
Sandip Mazumder ◽  
Ankan Kumar
Keyword(s):  
Radiative Transfer ◽  
Finite Volume ◽  
Unstructured Mesh ◽  
Laminar Flame ◽  
Cell Group ◽  
Benchmark Problems ◽  
Reacting Flow ◽  
Computational Domain ◽  
Participating Media ◽  
Cpu Time

Abstract The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the case without radiation.

Preconditioning and Solver Optimization Ideas for Radiative Transfer

2005 ◽  
Author(s):  
David B. Carrington ◽  
Vincent A. Mousseau
Keyword(s):  
Radiative Transfer ◽  
Search Algorithm ◽  
Linear Equations ◽  
Computational Domain ◽  
System Of Linear Equations ◽  
Participating Media ◽  
Preconditioning Techniques ◽  
Radiation Diffusion ◽  
Diagonal Scaling ◽  
Optimization Search

In this paper, radiative transfer and time-dependent transport of radiation energy in participating media are modeled using a first-order spherical harmonics method (P1) and radiation diffusion. Partial differential equations for P1 and radiation diffusion are discretized by a variational form of the equations using support operators. Choices made in the discretization result in a symmetric positive definite (SPD) system of linear equations. Modeling multidimensional domains with complex geometries requires a very large system of linear equations with 10s of millions of elements. The computational domain is decomposed into a large number of subdomains that are solved on separate processors resulting in a massively parallel application. The linear system of equations is solved with a preconditioned conjugate gradient method. Various preconditioning techniques are compared in this study. Simple preconditioning techniques include: diagonal scaling, Symmetric Successive Over Relaxation (SSOR), and block Jacobi with SSOR as the block solver. Also, a two-grid multigrid-V-cycle method with aggressive coarsening is explored for use in the problems presented. Results show that depending on the test problem, simple preconditioners are effective, but the more complicated preconditioners such as an algebraic multigrid or the geometric multigrid are most efficient, particularly for larger problems and longer simulations. Optimal preconditioning varies depending on the problem and on how the physical processes evolve in time. For the insitu preconditioning techniques—SSOR and block Jacobi—a fuzzy controller can determine the optimal reconditioning process. Discussions of the current knowledge-based controller, an optimization search algorithm, are presented. Discussions of how this search algorithm can be incorporated into the development of data-driven controller incorporating clustering and subsequent construction of the fuzzy model from partitions are also discussed.

scholarly journals A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment

2021 ◽  
Vol 13 (14) ◽  
pp. 7955
Author(s):  
Yongde Kang ◽  
Jingming Hou ◽  
Yu Tong ◽  
Baoshan Shi
Keyword(s):  
Numerical Model ◽  
Debris Flow ◽  
Finite Volume ◽  
Graphics Processing Unit ◽  
Morphological Changes ◽  
Finite Volume Scheme ◽  
Processing Unit ◽  
Algebraic Equations ◽  
Flow Process ◽  
Computing Technique

Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological processes was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the framework of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution of erosion and deposition over time. The research results provide guidance and a reference for the in-depth study of debris flow processes and disaster prevention and mitigation.

Reducing Undesired Wave Reflection at Domain Boundaries in 3D Finite Volume-Based Flow Simulations via Forcing Zones

2020 ◽  
Vol 64 (01) ◽  
pp. 23-47
Author(s):  
Robinson Peric ◽  
Moustafa Abdel-Maksoud
Keyword(s):  
Finite Volume ◽  
Wave Reflection ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Special Focus ◽  
Computational Domain ◽  
Wave Reflections ◽  
Flow Simulations ◽  
Large Eddy ◽  
Domain Boundaries

This article reviews different types of forcing zones (sponge layers, damping zones, relaxation zones, etc.) as used in finite volume-based flow simulations to reduce undesired wave reflections at domain boundaries, with special focus on the case of strongly reflecting bodies subjected to long-crested incidence waves. Limitations and possible sources of errors are discussed. A novel forcing-zone arrangement is presented and validated via three-dimensional (3D) flow simulations. Furthermore, a recently published theory for predicting the forcing-zone behavior was investigated with regard to its relevance for practical 3D hydrodynamics problems. It was found that the theory can be used to optimally tune the case-dependent parameters of the forcing zones before running the simulations. 1. Introduction Wave reflections at the boundaries of the computational domain can cause significant errors in flow simulations, and must therefore be reduced. In contrast to boundary element codes, where much progress in this respect has been made decades ago (see e.g., Clement 1996; Grilli &Horillo 1997), for finite volume-based flow solvers, there are many unresolved questions, especially:How to reliably reduce reflections and disturbances from the domain boundaries?How to predict the amount of undesired wave reflection before running the simulation? This work aims to provide further insight to these questions for flow simulations based on Navier-Stokes-type equations (Reynolds-averaged Navier-Stokes, Euler equations, Large Eddy Simulations, etc.), when using forcing zones to reduce undesired reflections. The term "forcing zones" is used here to describe approaches that gradually force the solution in the vicinity of the boundary towards some reference solution, as described in Section 2; some examples are absorbing layers, sponge layers, damping zones, relaxation zones, or the Euler overlay method (Mayer et al. 1998; Park et al. 1999; Chen et al. 2006; Choi &Yoon 2009; Jacobsen et al. 2012; Kimet al. 2012; Schmitt & Elsaesser 2015; Perić & Abdel-Maksoud 2016a; Vukčević et al. 2016).

scholarly journals A Finite-Volume Solution with a Bidirectional Upwind Difference Scheme for the Three-Dimensional Radiative Transfer Equation

2007 ◽  
Vol 64 (11) ◽  
pp. 4098-4112 ◽  
Author(s):  
Haruma Ishida ◽  
Shoji Asano
Keyword(s):  
Radiative Transfer ◽  
Difference Scheme ◽  
Finite Volume ◽  
Spherical Harmonic ◽  
Transfer Equation ◽  
Three Dimensional ◽  
Harmonic Series ◽  
Simultaneous Linear Equation ◽  
Upwind Difference Scheme ◽  
Cloud Models

Abstract A new calculation scheme is proposed for the explicitly discretized solution of the three-dimensional (3D) radiation transfer equation (RTE) for inhomogeneous atmospheres. To separate the independent variables involved in the 3D RTE approach, the spherical harmonic series expansion was used to discretize the terms, depending on the direction of the radiance, and the finite-volume method was applied to discretize the terms, depending on the spatial coordinates. A bidirectional upwind difference scheme, which is a specialized scheme for the discretization of the partial differential terms in the spherical harmonic-transformed RTE, was developed to make the equation determinate. The 3D RTE can be formulated as a simultaneous linear equation, which is expressed in the form of a vector–matrix equation with a sparse matrix. The successive overrelaxation method was applied to solve this equation. Radiative transfer calculations of the solar radiation in two-dimensional cloud models have shown that this method can properly simulate the radiation field in inhomogeneous clouds. A comparison of the results obtained using this method with those using the Monte Carlo method shows reasonable agreement for the upward flux, the total downward flux, and the intensities of radiance.

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